Any web site should have a single database. An unlimited number of tables can be added to any database.
Based on the information you have provided up to this point, your data is not normalized and you are manually creating new tables for each new class and major period of time. This is not general-purpose, reusable, and is not using the computer as a tool to accomplish a task. You have taken an activity done in the past using paper records and digitized it, but you are still spending time creating, moving, and maintaining the pieces of digital information yourself.
You would need to provide a specific example of what you mean and why you think there would be a problem.
If you have successfully designed your database and code, all this would involve is inserting a new row in a ‘course’ table, which would then create a new course_id value, then insert new rows in a course_student table, with the new course_id and the student_id values for the students registered for that course. This would then create new course_student_id values that would be used to recorded any data related to a specific student in a specific course.
Any code dealing with displaying or inserting data for any course should then ‘automatically’ adapt to the new course, simply based on the existence of the record being in the ‘course’ table.